inoculation strategies for victims of viruses
DESCRIPTION
TRANSCRIPT
![Page 1: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/1.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Inoculation Strategies for Victims of Viruses and the Sum-of-Squares Partition Problem
James Aspnes, Kevin Chang, and Aleksandr Yampolskiy
(Yale University)
![Page 2: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/2.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
ØMotivationnOur ModelnNash StrategiesnOptimal StrategiesnSum-of-Squares Partition ProblemnConclusion
![Page 3: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/3.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Question: Will you install anti-virus software?
Norton AntiVirus 2005 = $49.95
Value of your data = $350.00
Infection probability = 1/10
Expected loss = $35.00
![Page 4: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/4.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Answer: Probably not.
Norton AntiVirus 2005 = $49.95
Value of your data = $350.00
Infection probability = 1/10
Expected loss = $35.00
![Page 5: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/5.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
This selfish behavior…n …fails to achieve the social optimum.
![Page 6: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/6.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
What if instead…n …a benevolent dictator decided which
computers install an anti-virus?
Center node must install an anti-virus
or else!
![Page 7: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/7.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
nMotivationØOur ModelnNash StrategiesnOptimal StrategiesnSum-of-Squares Partition ProblemnConclusion
![Page 8: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/8.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model
n The network is an undirected graph G = (V,E).
n Installing anti-virus software is a single round non-cooperative game.
n The players are the network nodes: V = {0,1,…,n-1}.
![Page 9: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/9.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model : Strategies
n Each node has two actions: do nothing or inoculate itself.
n Strategy profile summarizes players’ choices.
n ai = probability that node i installs anti-virus software
![Page 10: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/10.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model : Attack Model
n After the nodes choose their strategies, the adversary picks a starting point for infection uniformly at random
n Node i gets infected if it has no anti-virus software installed and if any of its neighbors become infected.
![Page 11: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/11.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
0
2
1
Our Model : Attack Model (cont.)
3
54
n Example: Only node 3 installs anti-virus software. Adversary chooses to infect node 2.
![Page 12: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/12.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model : Attack Graph
0 1
2 3
54
0
2 3
54
network graph G attack graph Ga= G - Ia
1
![Page 13: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/13.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model : Individual Costs
n Anti-virus software costs C. Expected loss from virus is L.
n Cost of strategy to node i:
n Here, pi(a) = Pr[i is infected | i does not install an anti-virus]
![Page 14: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/14.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Our Model : Social Cost
n Social cost of is simply a sum of individual costs:
![Page 15: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/15.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
nMotivationnOur ModelØNash StrategiesnOptimal StrategiesnSum-of-Squares Partition ProblemnConclusion
![Page 16: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/16.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies
n Def: Strategy profile is in Nash equilibrium if no node can improve its payoff by switching to a different strategy:
for i = 0,...,n-1 and any x 2 [0,1],
n Fact: Nash strategies do not optimize total social cost (cf. Prisoner’s Dilemma)
![Page 17: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/17.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)
Thm: There is a threshold t=Cn/L such that each node in a Nash equilibrium¨ will install an anti-virus if it would otherwise end up in
a component of expected size > t¨ will not install an anti-virus if it would end up in a
component of expected size < t.¨ is indifferent between installing and not installing
when the expected size = t.
![Page 18: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/18.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)
n Corollary: Let t = Cn/L. Then a pure strategy is a Nash equilibrium if and only if¨Every component in Ga has size · t¨ Inserting any secure node j and its edges into
Ga yields a component of size ¸ t.
![Page 19: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/19.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)n Example: Let C=0.5,L=1 so that t=Cn/L=2.5.
Then is not a Nash equilibrium.
0 1
2 3
54
0
2 3
54
network graph G attack graph Ga= G - Ia
1
![Page 20: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/20.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)Thm: It is NP-hard to compute a pure Nash
equilibrium with lowest (resp., highest) cost.Proof sketch: By reduction to VERTEX COVER
(resp., INDEPENDENT DOMINATING SET).¨ Set C, L so that t=Cn/L=1.5. ¨ In a Nash equilibrium, (a) every vulnerable node
has all neighbors secure; (b) every secure node has an insecure neighbor
![Page 21: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/21.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)
n If V’µ V is a minimal vertex cover, then installing software on its nodes satisfies (a) because V’ is a vertex cover and (b) because V’ is minimal.
n Conversely, if V’ are secure nodes in a Nash equilibrium, then V’ is a vertex cover by (a).
![Page 22: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/22.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)
n Nash Theorem guarantees our game has a mixed Nash equilibrium.
n But does it make sense talking about pureNash equilibria?
![Page 23: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/23.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Nash Strategies (cont.)
Yes, it does!
Thm: If at each step some node with suboptimal strategy switches its strategy, the system converges to a pure Nash equilibrium in · 2n steps.
![Page 24: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/24.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy [KP99]n Price of anarchy measures how far away a
Nash equilibrium can be from the social optimum
n Formally, it is the worst-case ratio between cost of Nash equilibrium and cost of social optimum
n For network G and costs C, L, we denote it:
![Page 25: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/25.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy (cont.)Lower Bound: For a star graph K1,n,
ρ(G, C, L) = n/2.Upper Bound: For any graph G and any C, L,
ρ(G, C, L)· n.
Thm: Price of anarchy in our game is ρ(G, C, L) = Θ(n).
![Page 26: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/26.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy (cont.)Proof for lower bound:Consider a star graph K1,n. Let C=L(n-1)/n so that t=Cn/L=n-1.
G = K1,n
0
n-11
2
3n-2
…
![Page 27: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/27.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy (cont.)
Then, is an optimum strategy with cost C+L(n-1)/n.
G = K1,n
0
n-11
2
3n-2
…
Ga*
0
n-11
2
3n-2
…
![Page 28: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/28.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy (cont.)
And is worst-cost Nash with cost C+L(n-1)2/n.
G = K1,n
0
n-11
2
3n-2
…
Ga*
0
n-11
2
3n-2
…
![Page 29: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/29.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Price of Anarchy (cont.)
n Therefore,
n Proof for upper bound uses similar ideas.
![Page 30: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/30.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
nMotivationnOur ModelnNash StrategiesØOptimal StrategiesnSum-of-Squares Partition ProblemnConclusion
![Page 31: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/31.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Optimal Strategies
n So, allowing users to selfishly choose whether or not to install anti-virus software may be very inefficient
n Instead, let’s have a benevolent dictatorcompute and impose a solution maximizing overall welfare
![Page 32: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/32.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Optimal Strategies (cont.)
n We can show:Thm: Let t=Cn/L. If is an optimum strategy, then every component in Ga has size · max(1, (t+1)/2).
n Unfortunately,Thm: It is NP-hard to compute an optimal strategy.
![Page 33: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/33.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Optimum Strategies (cont.)n Naturally, we consider approximating the
solution.
0 1
2 3
54
0 1
2 3
54
network graph G attack graph Ga=G - Ia
k1=2
k2=2
secure nodes
Ia
![Page 34: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/34.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Optimum Strategies (cont.)
n For pure strategy , we have:
we concentrate on this part
![Page 35: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/35.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
nMotivationnOur ModelnNash StrategiesnOptimal StrategiesØSum-of-Squares Partition ProblemnConclusion
![Page 36: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/36.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Sum-of-Squares Partition
n We guess that there are m=|Ia| secure nodes.
n Problem: By removing a set of at most m · n nodes, partition the graph into components H1, …, Hk such that ∑i |Hi|2 is minimum.
![Page 37: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/37.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Sum-of-Squares Partition (cont.)
Thm: We can find a set of O(log2 n)¢m nodes whose removal partitions the graph into components H1,…,Hk such that ∑i |Hi|2 · O(1)¢OPT.
Proof sketch: We use the Leighton-Rao sparse cut algorithm [LR99]. The approach is similar to greedy log n approximation algorithm for set cover. We repeatedly remove the node cut that gives the best per-node benefit.
![Page 38: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/38.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Outline
nMotivationnOur ModelnNash StrategiesnOptimal StrategiesnSum-of-Squares Partition ProblemØConclusion
![Page 39: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/39.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Conclusionn We proposed a simple game for modeling
containment of viruses in a network.n Nash equilibria of our game have a simple
characterization.n We showed that, in the worst case, they can be
far off from the optimal solution.n However, a near-optimal deployment of anti-
virus software can be computed by reduction to the sum-of-squares partition problem.
![Page 40: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/40.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Open Problems
n Introduce a discount (or taxation) mechanism into the system.
n Suppose nodes can lie about their level of security (or about who their neighbors are). How do we make truth-telling a dominant strategy?
n Consider a “smart” adversary who targets the biggest graph component.
n How do we evaluate what C and L are?n Is there an algorithm for the sum-of-squares partition
problem with a better approximation ratio?
![Page 41: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/41.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Acknowledgments
Joan Feigenbaum, Hong Jiang, and YangRichard Yang
![Page 42: Inoculation strategies for victims of viruses](https://reader034.vdocuments.site/reader034/viewer/2022051513/546360a0b4af9f671c8b4bd9/html5/thumbnails/42.jpg)
Copyright (C) 2005 by Aleksandr Yampolskiy
Thank you!